121616949-math.212 - k has appropriate dimensions(namely...

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198 Chapter 9 Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | h | | | | | | 10 | | | | | 2 . . . . . . .. . . ....... . . . . . Figure 9.14 Cross-section of a conical water tank. A spring has a “natural length,” its length if nothing is stretching or compressing it. If the spring is either stretched or compressed the spring provides an opposing force; according to Hooke’s Law the magnitude of this force is proportional to the distance the spring has been stretched or compressed: F = kx . The constant of proportionality, k , of course depends on the spring. Note that x here represents the change in length from the natural length. EXAMPLE 9.16 Suppose k = 5 for a given spring that has a natural length of 0 . 1 meters. Suppose a force is applied that compresses the spring to length 0 . 08. What is the magnitude of the force? Assuming that the constant
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Unformatted text preview: k has appropriate dimensions (namely, kg/s 2 ), the force is 5(0 . 1-. 08) = 5(0 . 02) = 0 . 1 Newtons. EXAMPLE 9.17 How much work is done in compressing the spring in the previous ex-ample from its natural length to 0 . 08 meters? From 0 . 08 meters to 0 . 05 meters? How much work is done to stretch the spring from 0 . 1 meters to 0 . 15 meters? We can approximate the work by dividing the distance that the spring is compressed into small subintervals. Then the force exerted by the spring is approximately constant over the subinterval, so the work required to compress the spring from x i to x i +1 is approximately 5( x i-. 1)Δ x . The total work is approximately n-1 X i =0 5( x i-. 1)Δ x...
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